经济学课程003

经济学课程003

OntheUseofSymbolicComputationinUndergraduateMicroeconomicsInstructionDavidW.BoydEconomicanalysisandinstruction,atboththegraduateandundergraduatelevels,continuetoemploymoreandmoresophisticatedmathematicalmethods,yetthetechnologytypicallyusedbystudentstosolveeconomicsproblemsana-lyticallyhasnotadvancedmuchbeyondpencilandpaper.Inthelastdecadeorso,powerfulsoftwareprograms,generallycalledsymbolicprocessorsorcom-puteralgebrasystems,havebecomereadilyavailableforassistanceinsolvingproblemsmathematically.Ingeneralterms,symbolicprocessingprogramsarerelativelysophisticatedsoftwarepackagesthatcombinenumeric,symbolic,andgraphiccomputationinasingle,unifiedcomputingenvironment.Thepackagesalsoincludeprogrammingcapabilityformoreadvancedanalysis.Althoughtheprogramshavenotmadesubstantialinroadsintoeconomics,1mathematicians,scientists,andengineersareincreasinglyavailingthemselvesofthesepowerfultoolsandgeneratingasubstantialliteratureontheiruse.2Inthisarticle,Iillustratehowonesymbolicprocessingprogram,Maple,3caneasilybeincorporatedintoanadvancedundergraduatemicroeconomicscourse.IshowthatMaple’sgraphicalcapabilitiesprovideundergraduatestudentsinsightintomicroeconomicfunctionalrelationshipsbeyondthatpossibleinatextbookoronablackboard.Maplecanalsobeusedtohelpsolveanalyticallytheoftenalgebraicallytediousoptimizationproblemscommonlyfoundinadvancedun-dergraduatemicroeconomicscourses.Bylettingtheprogramperformthebulkofthemathematics,thestudentisfreetoconcentratemoreontheunderlyingeco-nomicsoftheproblemandontheeconomicintuitionbehindtheresults.Ifirstintroducesomeofthebasicsofsymbolicprocessing,layingafounda-tionforthespecificexamplesthatfollow.IthenprovideananalysisofhowMaplecanbeusedbothgraphicallyandanalyticallyinconsumertheory.SpecialattentionispaidtoMaple’scapabilitytodepictfundamentaleconomicrelation-shipsinthreedimensions.IconcludewithabriefdiscussionofmyexperiencesincorporatingMapleintoanundergraduatemathematicaleconomicscourse.DavidW.BoydisanassociateprofessorofeconomicsatDenisonUniversity(e-mail:boyd@deni-son.edu).ThisworkwassupportedinpartbytheFundfortheImprovementinPostsecondaryEdu-cation(FIPSE),GrantNo.P116B30079,andbyagrantfromtheW.M.KeckFoundation.TheauthorthanksZavenKarianandRonaldWintersforsharingtheirMapleexpertise,FrankHassebrockandRitaSnyderforhelpwithevaluationandassessment,andLauraBoydforhereditorialassistance.Thisversionhasbenefittedfromtheinsightfulcommentaryofthreeanonymousreferees.Summer1998227\nSYMBOLICPROCESSORS:THEBASICSTheintentofthisoverviewofMaple’sfunctionalcapabilities4isnottoofferinstructionontheintricaciesofMaplebutrathertoprovidebasicunderstandingofhowsymboliccomputationprogramswork.FormoredetailedinformationonMaple,seeChar(1993)orHeck(1993).Thenotationisasfollows:Maplecom-mandsmanuallyenteredbytheuserallbeginwiththesymbol>.CommandsendingwithasemicoloncauseMapletodisplayitsresponsedirectlyontothecomputer’smonitor.EndingacommandwithacolontellsMapletoperformtherequestedcalculationbutnottodisplaytheresultonthescreen.5Atthemostbasiclevel,Maple’snumericcapabilitiesallowittofunctionsim-ilarlytoasophisticatedcalculator.>4*5+7;27>ln(2);ln(2)>evalf(ln(2));.6931471806NoticethatMapletreatstheexpressionln(2)symbolicallyuntiltheuserexplicitlyrequeststhattheexpressionbeevaluatednumerically,withthecom-mandevalf.Thenumberofdigitsdisplayedintheresponsecanbeadjustedviaanoptiontotheevalfcommand.>evalf(ln(2),3);.693Acommonpracticeintheuseofsymbolicprocessorsistoassignanexpres-siontoavariable.ThisisaccomplishedinMaplewiththeoperator:=.>x:=7+sqrt(3);—x:=7+Ö3>evalf(x,5);8.7321Therealpowerofsymbolicprocessors,however,liesintheirabilitytomanip-ulateexpressionssymbolicallyandperformmathematicaloperationsonthoseexpressionssymbolically(suchasdifferentiation,accomplishedinMaplewiththecommanddiff).>z:=x^a*y^b;z:=xayb>diff(z,x);x(a–1)ayb228JOURNALOFECONOMICEDUCATION\nHere,thediffcommandoptiontellsMapletotakethepartialderivativewithrespecttothevariablex.>y:=1/3*x^3–3*x^2+8*x+7;132y:=x-3x+8x+73>dydx:=diff(y,x);dydx:=x2–6x+8>solve(dydx=0,x);4,2Inthefinalthreecommandslistedabove,Maplesolvestheequationdy/dx=0forx.ThisprocesscanbemadesomewhateasierbycombiningtwooftheMaplecommands,achievingthesameresult.>solve(diff(y,x)=0,x);4,2SYMBOLICCOMPUTATIONINUNDERGRADUATEMICROECONOMICSINSTRUCTION:ANEXAMPLEAstandardproblemfromconsumertheoryinmicroeconomicsisthemaxi-mizationofutilitysubjecttoabudgetconstraint.Thisproblemcanbesolvedgraphicallyintwodimensionsusingindifferencecurvesandabudgetline,oranalyticallyintwoorhigherdimensionsusingthemethodofLagrangemultipli-ers.Symbolicprocessingprogramscanprovideassistancetostudentsusingeitherorbothofthetwosolutionmethods.Inthegraphicalapproach,theplot-tingcapabilitiesofsymbolicprocessorscanprovideinsightintotheunderlyingutilityfunctionandwhatitimpliesaboutrelativepreferences.Iftheproblemissolvedanalytically,theabilitytomanipulateequationssymbolicallycanallowthestudenttofocusmoreontheunderlyingeconomiclogicoftheapproachandlessonthemathematicalcalculationofthesolutionsthemselves.IillustrateherehowMaplecanbeappliedbothgraphicallyandanalyticallytothisproblem.GraphicalToolsConsiderthesimpleandstandardsituationwherearepresentativeconsumermustallocateafixedandgivenbudgetacrossjusttwoconsumptiongoods,XandY.Thelevelofhappinessassociatedwithconsumptionofvariouscombinationsofthetwogoodsisrepresentedbytheindividual’sutilityfunction,U(X,Y).Manyleadingintermediatemicroeconomicstextbooksneverdisplayanexampleofsuchathree-dimensionalutilityfunction,potentiallyleavingstudentswiththemistakenimpressionthatthesolegraphicalrepresentationofutilityisanindif-ferencemap.6Symbolicprocessors,however,freestudentstoexaminegraphi-callyanynumberofutilityfunctions.Asoneexample,considerthespecificSummer1998229\nCobb-DouglasutilityfunctionU=X1/4Y3/4thatcaneasilybeassignedandplot-tedinthreedimensionsinMaple,usingtheplot3dcommand7(Figure1).>utility:=X^(1/2)*Y^(1/2);——utility:=ÖXÖY>plot3d(utility,X=0..10,Y=0..10,axes=boxed,orientation=[–133,65],title=`Cobb-Douglasutilityfunction`);Ofcourse,mostundergraduateeconomicsstudentsthinkofthegraphicalrep-resentationofutilityasindifferencecurves.Byalteringtheoptionstotheplot3dcommand,Maplecansuppressthethree-dimensionalutilityfunctiondiagramaboveintoaplotoftwo-dimensionalindifferencecurvessofamiliartoeconom-icsstudents(Figure2).>plot3d(utility,X=0..10,Y=0..10,style=contour,axes=normal,color=black,orientation=[–90,0],title=`Cobb-Douglasindif-ferencecurves`);Thepowerofsymboliccomputationenablesstudentstoseeforthemselvesthatindifferencecurvesare,infact,nothingmorethancurvesoflevelutility.InMaple,ausercansimplyselectthediagramabovewiththecomputer’spointingdevice,ormouse,andadjusttheplotorientationmanuallybymovingthemouseuntilthediagramexpandstothreedimensions.SuchasimpleactionmightresultintheplotofFigure3.Thesetypesofinteractiveexercisesallowstudentstoseethedirectrelation-shipbetweenathree-dimensionalutilityfunctionanditsassociatedtwo-dimen-sionalindifferencecurves.Moreover,astudentisfreetoviewanytwo-orthree-dimensionalplotwithanyorientationheorshedesires,inessencecreatingaFIGURE1Cobb-DouglasUtilityFunction230JOURNALOFECONOMICEDUCATION\nFIGURE2Cobb-DouglasIndifferenceCurvesFIGURE3Cobb-DouglasIndifferenceCurvespersonalizedperspective.Ofcourse,thesetypesof“whatif”experimentswouldnotbepossiblewitheitherpencilandpaperorachalkboard.Maple’splottingcapabilitiesalsoallowstudentstoexaminegraphicallytheconsequencesofchangesinrelativepreferencesbetweenthetwogoodsXandYonboththeutilityfunctionandtheindifferencecurves.Specifically,Maplecon-tainsananimationfeaturethatgeneratesaseriesofrelatedplots.Asanexample,considerthemoregeneralCobb-DouglasutilityfunctionU(X,Y)=XaY1–a,where,inlooseterms,theexponentacapturesthedegreeofrelativepreferenceforgoodSummer1998231\nXovergoodY.Inthesequencebelow,theMaplecommandanimate3dpro-ducesaseriesofplotsthatshowhowtheutilityfunctionchangesastheexponentamovesfrom0.2to0.8ineightequalincrements.8>utility:=X^(a)*Y^(1-a);utility:=XaY(1–a)>animate3d(utility,X=0..10,Y=0..10,a=0.2..0.8axes=boxed,orientation=[–133,65],style=wireframe);Oncetheplotsaregeneratedbytheprogram,theuserhastheoptionofview-ingthesequenceinorder,likeashortmotionpicture,orexaminingeachseparatediagramindividually.Thefirstandlastoftheseriesofeightplotsgeneratedbytheanimate3dcommandareshowninFigure4.FIGURE4UtilityFunctionAnimation232JOURNALOFECONOMICEDUCATION\nIfviewedtogetherasasequence,thethree-dimensionalutilitysurfaceappearssomewhatlikeanundulatingwavemovingfromtheXaxistowardtheYaxis,illustratingtostudentshowrelativepreferencesbetweenthetwogoodsXandYchangeastheexponentavaries.Animationcanalsobeappliedtoindifferencecurvesbyusingthefollowinganimate3dcommand:>animate3d(utility,X=0..10,Y=0..10,a=0.2..0.8,axes=normal,orientation=[–90,0],style=contour);ThefirstandlastoftheseriesofeightplotsgeneratedbythiscommandareshowninFigure5.Ifviewedtogetherasasequence,theindifferencecurvesslowlybecomemoreFIGURE5IndifferenceCurveAnimationSummer1998233\nsteep,asrelativepreferencesmoveawayfromgoodYtowardgoodX,againillus-tratingtostudentstheeconomicimportanceoftheexponentaintheutilityfunction.SolvingtheConstrainedOptimizationProblemAnalyticallyInthissection,Maplewillbeusedtosolveanalyticallytheproblemofmaxi-mizingutilitysubjecttoabudgetconstraintbyusingthemethodofLagrangemultipliers.Again,letusconstrainourselvestotheworldofjusttwogoods,XandY.LetpreferencesoverthesetwogoodsbegivenbythegeneralCobb-Dou-glasutilityfunctionU(X,Y)=XaYb.DenotethepricesofgoodXandgoodYbyPandP.Theconsumer’sincomepermarketperiodisgivenbyInc.9xyFirst,thisinformationisenteredintoMaple.>utility:=X^a*Y^b;utility:=XaYb>constraint:=Inc–Px*X–Py*Y;constraint:=Inc–PxX–PyYNexttheLagrangianexpressionisentered,withlastheLagrangemultiplier.>maxutility:=utility+lambda*constraint;maxutility:=XaYb=l(Inc–PxX–PyY)Tomaximizethisexpression,first-orderconditionsinvolvetakingpartialderiva-tivesofmaxutilitywithrespecttoX,Y,andl.Recallthatpartialdifferentia-tionisperformedinMaplewiththediffcommand.>FOC1:=diff(maxutility,X);abXaYFOC1:=-lPxX>FOC2:=diff(maxutility,Y);XaYbbFOC2:=-lPyY>FOC3:=diff(maxutility,lambda);FOC3:=Inc–PxX–PyYTypically,themostdifficultpartoftheproblemforstudentsistosetthesefirst-orderconditionsequaltozeroandsolvesimultaneouslyfortheoptimalvaluesofX,Y,andl.Becausemostpracticalexamplesofutilityfunctionsarenonlinear,thesystemofthreeequationsinthreeunknownsinvolvestwononlinearequa-tions,whichcanbeparticularlyintractable.Symboliccomputationminimizesthisproblembysolvingthesystemofsimultaneousequationsanalytically.InMaple,thisisaccomplishedinasinglestatementwiththesolvecommand.Here,weassigntheproblemsolutionstoavariablecalledsolutionset.>solutionset:=solve({FOC1=0,FOC2=0,FOC3=0},{X,Y,lambda});234JOURNALOFECONOMICEDUCATION\nSolutionset:=abæIncaöæbIncöç÷ç÷(a+b)bIncè(a+b)PxøèPy(a+b)øInca{Y=,l=,X=}Py(a+b)Inc(a+b)PxAlthoughatthispointMaplehassolvedtheproblemfortheoptimalvaluesofX,Y,andl,thesolutionsthemselveshavenotbeenassignedtothevariables.Assignmentsareaccomplishedwiththeassigncommand>assign(solutionset);Oncetheassignmentofthesolutionstothevariableshasbeenmade,wecanaskMapletodisplaytheresults.>X:=X;Y:=Y;IncabIncX:=Y:=(a+b)PxPy(a+b)ThesolutionstoXandYare,ofcourse,thefamiliarordinary,orMarshallian,demandfunctions.Studentscanseedirectlyfromtheseexpressionsthat,givenpositiveaandb,10bothMarshalliandemandcurvesaredownwardsloping,thatbothcommoditiesarenormalgoods,andthatthetwogoodsareneithergrosssubstitutesnorgrosscomplements.Morespecifically,withalittlemanipulation,MaplecanplottheMarshalliandemandscheduleforgoodX.First,topresentthegraphintheformfamiliartoeconomicsstudentswithpriceontheverticalaxis,weneedtosolvetheexpres-sionforXexplicitlyforPx.Temporarydummyvariables,Px1andX1,areintro-ducedtopreservetheoriginals,PxandX,asvariables.>Px1:=solve(X=X1,Px);IncaPx1:=-X1a-X1bAsisoftenthecase,alittlecoercionisnecessarytogetMapletomakethedenominatorlookmorepalatable.ThisisaccomplishedwiththeMaplecom-mandfactor,whichfactorsthedenominatorofthefraction.>Px1:=factor(Px1);IncaPx1:=X1(a+b)Tomaketheplot,wemustsubstituteinnumericalvaluesforthevariablesa,b,andInc.Substitutionisaccomplishedwiththecommandsubs.>Px1:=subs(a=1/4,b=3/4,Inc=100,Px1);Summer1998235\n1Px1:=25X1Now,theMarshalliandemandcurvecanbeplotted,asinFigure6,overthearbi-traryrangeof10to20forthevariableX1.>plot(Px1,X1=10..20,title=`MarshalliandemandforX`);ToshowstudentsthatgoodXisnormal,ratherthaninferior,wecouldleaveIncasavariable,andMaplecouldanimateoverarangeofInc.AsIncincreased,thedemandcurveshowninFigure6wouldshiftoutward.TheoptimalvalueoftheLagrangianmultiplier,l,isalsoofeconomicinter-est.ItcanbeexpressedinamoresimplifiedversionthanthatinsolutionsetbyusingtheMaplecommandsimplify.>lambda:=simplify(lambda,power,symbolic);l:=Inc(a+b–1)aa(a+b)(–a–b+1)Px(–a)bbPy(–b)Theeconomicinterpretationoflisthemarginalutilityofonemoredollarofincome.Fromtheaboveexpression,studentscanseethatlvariesinverselywithprices.Thismakeseconomicsense—higherpricesmeanonemoredollarofincomegivesyoulessadditionalhappinessorsatisfaction.Furthermore,therela-tionshipbetweenlandincomedependsonthesizeoftheexponentsaandb.Ifa+b>1,themarginalutilityofonemoredollarofincomeincreaseswithris-ingincomelevels,whereastheconditiona+b<1indicatesdiminishingmar-ginalutilityofincome.Iftheexponentssumexactlytoone,themarginalutilityofonemoredollarofincomeisindependentofincomelevel.Aseriesofsubstitutionsandpartialderivativesmayhelpstudentsseetheseresultsmoreclearly.Considerfirstthecasewherea+b>1.SpecificvaluesforFIGURE6MarshallianDemandforX236JOURNALOFECONOMICEDUCATION\naandbthatsatisfythisconditioncanbesubstitutedintoatemporarydummyvariable,l1.>lambda1:=subs(a=1,b=1,lambda);1Incl1:=2PxPyWecannowpartiallydifferentiatel1withrespecttoInc,Px,orPy.TheMaplecommandDiffisusefulhere.WhereasdiffcausesMapletocalculatethepartialderivative,theinertDiffcommandinstructsMapletoleavethepartialderivativeasasymbolfordisplay.>Diff(‘lambda1’,Inc)=diff(lambda1,Inc);¶11l1=¶Inc2PxPy>Diff(‘lambda1’,Px)=diff(lambda1,Px);¶1Incl1=-¶Px2Px2PyThefirstpartialderivativeispositive,whereasthesecondisclearlynegative,indicatingthat,whena+b>1,themarginalutilityofincomevariesdirectlywithincomeandnegativelywithprices.Asimilarsetofsubstitutionsandpartialderivativescouldalsobeperformedfortheremainingtwocasesofa+b<1anda+b=1,ifnecessary.Finally,theindirectutilityfunctioncanbeobtainedbysubstitutingtheoptimalvaluesforXandYbackintotheutilityfunction,U(X,Y)=XaYb.InMaple,becausethevariablesXandYhavebeenassignedtheiroptimalvalues,theopti-malXandYhavealreadybeensubstitutedintoanyexpressioninwhichtheyappear.Thus,theindirectutilityfunctioncanbeobtainedbymerelyaskingMapletodisplayutility.>utility:=utility;utility:=Inc(a+b)aa(a+b)(–a–b)Px(–a)bbPy(–b)Asubstitutionfollowedbysomepartialderivativessimilartotheonesabovemightproveusefulhere.>utility1:=subs(a=1/2,b=1/2,utility);1Incutility1=2PxPy>Diff(‘utility1’,Inc)=diff(utility1,Inc);¶11utility1=¶Inc2PxPySummer1998237\n>Diff(‘utility1’,Py)=diff(utility1,Py);¶1Incutility1=-¶Inc4PxPy3/2Asintuitionwouldsuggest,themaximumlevelofattainableutilityvariesdirectlywithincomeandinverselywithprices.Indeed,aprimarybenefitofusingsymboliccomputationisthatstudentscandevotemoreeffortandenergytounderstandingtheeconomicintuitionbehindtheirresults,ratherthancon-centratingdisproportionatelyonthemathematicalcalculationofthesolutionsthemselves.FurtherExtensionswithSymbolicComputationUndergraduatestudentsstudyingmicroeconomicsarefamiliarwiththetwo-dimensionalgraphicalrepresentationoftheconsumer’sproblemthatusesbudgetlinesandindifferencecurves.Asshownabove,theplottingcapabilitiesofsym-bolicprocessorsallowstudentstoexploreinthreedimensionstheutilityfunctionthatgeneratesthetwo-dimensionalindifferencecurves.Three-dimensionalplot-tingcanprovideadditionalinsightintothisproblem.Inthissection,Maple’ssymbolicprocessingandgraphicaltoolsareusedtoextendtheconsumer’sprob-lemtoincludethreeconsumptiongoods.Thisextensionshouldillustratethat,countertosolvingtheproblembyhand,expandingtheproblemtoincludemoregoodsdoesnotmakethesolutionanymoredifficultifasymbolicprocessorisused.Theprogramstatementsaresubstantivelyunchanged.Furthermore,thegraphicaltoolsofMaplecanclearlydepictthethree-dimensionalanalogofbud-getlinesandindifferencecurves.Webeginwitharepresentativeconsumer’spreferencesoverthreeconsump-tiongoods,X,Y,andZ,ascapturedbythespecificutilityfunction,U(X,Y)=X1/2Y1/2Z1/2.Letincomebe$120permarketperiodandthepriceofthethreegoodsbenormalizedtounity.First,thisinformationisenteredintoMaple,andtheconstrainedutilitymaximizationproblemisformed.11>U:=X^(1/2)*Y^(1/2)*Z^(1/2):>constraint:=120–X–Y–Z:>maxU:=U+lambda*constraint;maxU:=XYZ+l(120-X-Y-Z)Thenthefirst-orderconditionsaretaken.>FOC1:=diff(maxU,X):>FOC2:=diff(maxU,Y):>FOC3:=diff(maxU,Z):>FOC4:=diff(maxU,lambda):238JOURNALOFECONOMICEDUCATION\nNext,Maplesolvesthissystemoffourequationsfortheoptimalvaluesofthedecisionvariables,X,Y,Z,andl.Then,thesolutionsareassignedtothevari-ables.>solutions:=solve({FOC1=0,FOC2=0,FOC3=0,FOC4=0},{X,Y,Z,lambda}):>assign(solutions):Giventhesymmetryoftheutilityfunctionandequalprices,theoptimalvaluesofX,Y,andZareallthesame.>X:=X;Y:=Y;Z:=Z;X:=40Y:=40Z:=40Theseoptimalvalueshavealreadybeensubstitutedbackintotheutilityfunction,U,yieldingthemaximumlevelofattainableutility,givenincomeandprices.>U:=U;U:=8010NowMaple’splottingcapabilitiescanbeusedtodepicttheconsumeropti-muminthreedimensions.Becauseweknowthelevelofoptimumutilityis—80Ö10,thethree-dimensionallevelsurfaceassociatedwiththisparticularquanti-tyofutilitycanbeplottedinX-Y-ZspaceneartheoptimumpointofX=Y=Z=40.12This,ofcourse,isthethree-dimensionalextensionofatwo-dimensionalindifferencecurve,whichmosteconomicsstudentshaveneverseen(Figure7).>Zutil:=solve(U=80*sqrt(10),Z);FIGURE7IndifferenceSurfaceSummer1998239\n1Zutil:=64000YX>plot3d(Zutil,X=30..50,Y=30..50,axes=boxed,orienta-tion=[–40,75]);Theorientationofthisdiagramisslightlydifferentfromthepreviousplots,per-mittingtheindifferencesurfacetobeviewedfromitsside.Withthreeconsumptiongoods,thebudgetconstraintisatwo-dimensionalplaneinX-Y-Zspace.ThiscanalsobeplottedinMaple,asinFigure8.>Zinc:=120–X-Y;Zinc:=120–X–Y>plot3d(Zinc,X=30..50,Y=30..50,axes=boxed,orienta-tion=[–40,75]);Thetwoplotscanalsobecombinedinasinglediagram(Figure9)thatillus-tratesthetangencybetweenthethree-dimensionalutilitysurfaceassociated—withtheoptimallevelofutility(80Ö10)andthebudgetplane.>plot3d({Zutil,Zinc},X=30..50,Y=30..50,axes=boxed,orienta-tion=[–40,75]);Ofcourse,inMaple,studentsarefreetoselecttheabovediagramwiththecur-sorandaltertheorientationoftheplottoviewthetangencyfromdifferentper-spectives.Forexample,itmightmakesensetoexaminethetwosurfacesatanangledirectlyorthogonaltothepointoftangency(Figure10).FIGURE8BudgetPlane240JOURNALOFECONOMICEDUCATION\nFIGURE9IndifferenceSurfaceTangenttoBudgetPlaneFIGURE10IndifferenceSurfaceTangenttoBudgetPlane—SideViewEVALUATIONANDASSESSMENTInthespringsemesterofthe1994–95academicyear,IincorporatedMapleintomyundergraduatemathematicaleconomicsclassatDenisonUniversity.Pre-requisitesforthiscourseareintermediatemicroeconomicanalysisandthefirstcourseinatwo-semestercalculussequenceofferedbythemathematicsdepart-ment.InDenison’sDepartmentofEconomicslaboratory-basedcurriculumSummer1998241\n(KingandBartlett1990;orKingandLaRoe1991),allintermediatecoursesandmostupper-levelcoursescombinetraditionallectureperiodswithaonce-week-lytwo-hourlaboratorysession.Mathematicaleconomicsisoneoftheadvancedcoursesthatincludesacomputerlaboratorycomponent.13Tenstudents(alleconomicsmajors)wereenrolledinmathematicaleconom-icsduringthespring1995semester,afiguresomewhatbelowtheaverageforthepreviousthreeyears.Sixofthesestudentswerejuniors;theremaining4wereseniors,2ofwhomhadbeenacceptedintoPh.D.economicsprograms.ThestudentshadameanSATcombinedscoreof1090andanaverageSATmathematicsscoreof601.14Themeangradepointaverageforthestudentsafterthespring1995semesterwas3.09,ona4.0scale.SixofthestudentshadnopriorexposuretoMaple.Theintermediatemicroeconomicanalysisprereq-uisiteassuredthatallstudentswere,however,comfortableworkingwithaMacintoshcomputer.Duringthesemester,weusedMapleatotaloffivetimesinstructuredclassset-tings.15TwiceIconductedclasslecturesinthecomputerlaboratory,demonstrat-ingviaanoverheadprojectiondevicehowMaplecouldanalyzeproblemswehadpreviouslysolvedbyhand.TheseclassesgavestudentsageneralideaofhowsymbolicprocessingworksandhowmathematicalstepscouldbetranslatedintospecificMaplecommands.Inaddition,studentsusedMapleinthreelaboratorysessions.16ThefirstofthesewasessentiallyanintroductiontoMaple,duringwhichstudentsweregiventheirfirstopportunitytoworkwiththeprogram.Towardtheendofthislaboratory,studentswereaskedtouseMapletoperformsomebasicmathematicsexercises,includingtakingsimplepartialderivativesandsolvingsystemsofequationsforsolutions.Thisonetwo-hoursession,incombinationwithabriefeight-pagehandout,wassufficienttoimparttothestu-dentsabasicworkingknowledgeofMaple.Intheremainingtwolaboratoryses-sions,studentsworkedindividuallyorinsmallgroupstosolveavarietyofmath-ematicaleconomicsproblems,similartothetypesofexercisesdescribedearlierinthisarticle.StudentsalsousedMapleoutsideofclasstohelpthemcompletehomeworkassignmentsandtoprepareforexaminations.Theend-of-semesterstudentsurvey,containingbothquantitativeandquali-tativequestions,indicatedthatthestudentshadagenerallyfavorableexperi-encewithsymboliccomputation.Eightofthe10questionnairescharacterizedtheexperiencewithMapleashelpfulorpositive.Anumberofstudentscom-mentedthatMapleallowedthemtofocusmoreoneconomicsandlessonmath-ematics.Mapleisaquickerwayofcomputingdifficultandtediousmathproblems.Ifany-thing,ithelpedwithmyunderstandingofthefinalproductwhich,withoutMaple,itmighthavetakenquiteawhiletogetto.(emphasisinoriginal)Maplecutdownonthebusyworkofcomputingandallowedmetofocusontheeco-nomicprinciplesbehindit.Itwashelpfulbecauseitmadeiteasiertotakederivatives...Commentaryalsorevealedthatmanystudentsfeltthatactuallyseeingeco-nomicrelationshipsinthreedimensionswasquitehelpful.Moreover,somestu-242JOURNALOFECONOMICEDUCATION\ndentsindicatedthatMaplewasacomplementto,ratherthanasubstitutefor,thepencil-and-paperwork.Tothem,Maplewasmoreusefulasamethodofcheck-inghandcalculations,asopposedtoastand-alonesolver.17Overthecourseofthe14-weeksemester,studentsimprovedtheirabilitytoworkwithMaple.WhenaskedattheendofthesemesterhowgoodtheywereatusingMaple,ona7-pointscalewith1indicatingverypoorand7signifyingverygood,theclassaveragewas4.15.Notallstudentresponseswerepositive.Sevenofthe10studentscommentedthattheiruseofMaplewasconstrainedbythespeedandmemoryrequirementsofourlaboratorycomputers.Indeed,whenaskedabouttheirleveloffrustrationinusingMaple,ona7-pointscalewith1representingnotatallfrustratedand7signifyingveryfrustrated,theclassaveragewas4.4.Nonetheless,IbelievethatMapleimprovedthecourseinanumberofways.Assomeofthestudentcommentsaboveindicate,studentswereabletodevotemoretimeandenergytounderstandingtheeconomicintuitionbehindtheirresultsandlessonthemathematicalcalculationoftheanswersthemselves.Ialsofeltthatsimplybyseeingsomeoftheunderlyingeconomicrelationships(e.g.,utilityfunctionsandproductionfunctions),studentsdevelopedadeeperandmoreintuitivegraspoftheeconomicproblemsweweresolving.Onamoreconcretelevel,symbolicprocessingmadethenotionofordinalutil-itymucheasiertogetacrosstostudents.Ordinalutilityimplies,interalia,thatMarshalliandemandfunctionsareinvarianttomonotonictransformationsoftheutilityfunction.Becausethesolutiontotheconstrainedutilitymaximizationproblemissotime-consuming,itisoftendifficulttogetacrosstostudentsthatmonotonictransformationsofutilityyieldthesameMarshalliandemands.How-ever,whenusingasymbolicprocessor,oncetheprogramstatementstosolvetheconstrainedutilitymaximizationproblemhavebeendeveloped,itisastraight-forwardandalmostimmediateprocesstogobackandalterthegivenutilityfunc-tionandseetheresultofidenticalMarshalliandemands.Forexample,thefollowingseriesofMaplecommandsgeneratetheMarshal-liandemandsfromaspecificinputtedutilityfunction:>U:=X^(1/4)*Y^(3/4):>constraint:=Inc–Px*X–Py*Y:>maxU:=U+lambda*constraint:>FOC1:=diff(maxU,X):FOC2:=diff(maxU,Y):FOC1:=diff(maxU,lambda):>answers:=solve({FOC1=0,FOC2=0,FOC3=0},{X,Y,lambda}):>assign(answers):>X:=X;Y:=Y;1Inc3IncX:=Y:=4Px4PyAtthispoint,astudentcaneasilyreturntothefirstcommandandchangetheSummer1998243\nutilityfunctiontosomemonotonictransformationofU=X1/4Y3/4.Aftersimplyhittingthereturnorenterkeyaseriesoftimes,thestudentwillhaveineffectreworkedtheconstrainedoptimizationproblemforadifferentutilityfunctioninliterallyseconds.Indeed,itisaquitesimpleprocesstowriteaMapleprocedure,containingalltheabovecommands,whichtakesasinputautilityfunctionandreturnsasoutputtheMarshalliandemands.Shouldaninstructorwishtowritesuchaprocedure,studentswouldseeanimmediateone-to-onerelationshipbetweenautilityfunctionanditsassociatedMarshalliandemands.Forexample,ifsuchaprocedurecalleddemandswerewritten,theMapleinteractionwouldappearasfollows:>demands(X^(1/4)*Y^(3/4));1Inc3IncX:=Y:=.4Px4PyAsaninstructor,theinvestmentintimetolearnMapleandincorporateitintomyclasswasrelativelymodest.Overthesummerof1994,Iattendedatwo-weekseminarconductedbytwoDenisoncolleagues(amathematicianandaphysicist)aspartofagrantfromtheFundfortheImprovementinPostsecondaryEduca-tion(FIPSE)toimplementsymboliccomputationacrossawidearrayofdisci-plinesatDenison.ThosetwoweeksprovidedmorethanenoughtimetobecomesufficientlyproficientinMaple.Ididencounterseveralproblemswiththeprogram.ByfarthelargestobstacleIfacedwasthatourcomputerlaboratoryconfigurationwassimplyinadequateforMaple’ssubstantialhardwarerequirements.Onaverage,ittookstudentsroughly90secondstogenerateasimplethree-dimensionalplotontheircomput-ers.Studentcomputerstook6to8minutestoproducetheanimationsdescribedearlierinthearticle.Theinstructormachine,withitscoprocessor,roughlyhalvedthesetimes.Nevertheless,studentswereclearlyfrustratedandoftenrefrainedfromusingMaple’sgraphicalcapabilities.Althoughtheseperformanceissuescanbeadequatelyaddressedwithnewertechnology,18Ididencounterotherproblemsthatarenothardwarespecific.Forexample,MaplecannotsolvefortheMarshalliandemandsfrommorecomplextypesofutilityfunctionswhenthemethodofLagrangemultipliersisused.Ofcourse,othersolutionmethodsareavailable,buttheyarenotoftentaughtineconomicscourses.Moreover,Mapleoccasionallydisplayssolutionsinaformdifferentfromthatfamiliartoeconomists.AlthoughMaplecanusuallybecoercedintorearrangingfunctionalformsbyusingappropriatecommands,therearecertainlyinstancesinwhichtheuserisstuckwithanunfamiliarfor-mat.InstructorsusingMapleshouldbepreparedtoexpendsomeresourcesconvincingstudentsthattwoexpressionsareequivalenteventhoughtheydonotappearidenticalatfirstblush.Finally,studentperformanceonexamina-tionsdidnotsignificantlyimprovewiththeintroductionofsymbolicprocess-ing.Isuspect,however,thatthisisatleastinpartbecauseofmyinexperienceinformulatingquestionsthatdrawontheadditionalinsightgainedfromuseofsymbolicprocessors.244JOURNALOFECONOMICEDUCATION\nNOTES1.TheworkofHalVarian,includinghis1992bookinwhichheemployedMathematica(Varian1992a),isonenotableexception.2.See,forexample,Karian(1992aand1992b).3.Anumberofothersymbolicprocessorsarealsoavailableontoday’smarket,perhapsthemostwidelyusedofwhichisMathematica.MoreinformationonMathematicaisavailablefromtheMathematicahomepage,http://www.wri.com/.TheselectionofMapleoverMathematicaasthesymbolicprocessorofuseinthisarticlereflectssolelytheauthor’sfamiliarityandshouldnotbeconstruedtoimplythattheformerisinanywaysuperiortothelatter.Indeed,forvirtuallyalleconomicsapplicationsappropriateforundergraduates,thetwoaresubstantivelynearlyinterchangeable.OthersymbolicprocessorscurrentlyavailableincludeDerive,Reduce,andMacsyma.4.MapleisavailablefromWaterlooMapleSoftwareandrunsonawidevarietyofcomputingplat-forms.Formoreinformation,seeMaple’shomepage,http://www.maplesoft.com/.AfulllistofcomputingplatformsonwhichMapleiscapableofrunning,includingspecifichardwareandsoftwarerequirements,isprovidedathttp://www.maplesoft.com/Products/MapleV/MapleAvail.html.5.ItisalsopossibletodirectMapletowriteoutputtoaseparatefile.Thisisparticularlyusefulwithgraphicaloutput.6.Ofnineintermediateandadvancemicroeconomicstextbooks(BingerandHoffman1988;BrowningandBrowning1992;GouldandLazear1989;HendersonandQuandt1980;KatzandRosen1994;Nicholson1995;PindyckandRubinfeld1992;Varian1992b1993),onlyGouldandLazear(1989,41)attemptedtodepictathree-dimensionalutilitysurface.7.Mostoftheoptionstotheplot3dcommandarerelativelystraightforward,withthepossibleexceptionoftheorientationoption.Briefly,thisoptionisspecifiedasorientation=[q,f],whereqisthecounterclockwiserotationindegreesfromthepositivexaxistowardthepositiveyaxis,andfistherotationupwardtowardthepositivezaxis.Thedefaultpositionsareq=f=45º(seeChar1993,173).8.Thenumberofincrementscanbechangedusingtheframesoptiontotheanimate3dcom-mand.Thedefaultnumberofframesis8.—9.IncomeisdenotedbyIncratherthanIbecauseMaplereservesthenameIforÖ–1.10.Positiveexponentsarenecessaryifthecommoditiesinquestionaretobeeconomicgoods.11.Maple’soutputtomanyoftheseandothercommandsthatfollowhavebeensuppressedbyend-ingthestatementswithcolonsratherthansemicolons.12.Becausetheoptimalvalueof40hasalreadybeenassignedtothevariablesX,Y,andZ,theMaplecommandrestartmustbeex—ecutedfirst.Otherwise———,theutilityfunctionUwouldstillbeassigneditsspecificvalueofÖ40Ö40Ö40=80Ö10.13.Atthetime,theDepartmentofEconomicscomputerlaboratorywasequippedwith24AppleMacintosh®Centris610computers,eachwithacolormonitor,8megabytesofrandomaccessmemory(RAM),andan80megabyteharddrive,forstudentuse,and1AppleMacintosh®Cen-tris650computer,withacoprocessor,acolormonitor,8megabytesof(RAM),andan80megabyteharddrive,connectedtoanoverheaddisplaydevice,forinstructoruse.MapleVfortheMacintosh,Release3,wasinstalledonallthesecomputers.Forspecifichardwarerequire-mentsforthecurrentreleaseofMapleforuseontheMacintosh®orothercomputingplatforms,seehttp://www.maplesoft.com/Products/MapleV/MapleAvail.html.14.TheseaveragesreflecttheninestudentswhotooktheSATpriortoenrollmentatDenison.15.Ipurposelychosetolimitstudents’classtimeexposuretoMapletopreventanydependenceonsymbolicprocessingfromdeveloping.Becausemanyofthestudentsinthisclasshavehistori-callygoneontograduateprogramsineconomicsorbusiness,veryfewofwhichusesymboliccomputationintheircurricula,aprimaryobjectiveofthecourseisforstudentstolearntosolvethesetypesofproblemsbyhand.16.Ininstructionalsettingswithoutaccesstoacomputerlaboratory,thetypesofexercisesdescribedabovecouldbedemonstrateddirectlytostudentsinalectureroomequippedwithacomputerconnectedtoanoverheaddisplaydevice.Indeed,usingsymbolicprocessinginthisfashionmin-imizestheinvestmentinhardwareandsoftware,aswellasstudentlearningtime.Moreover,thegraphicalcapabilitiesofsymbolicprocessorscouldbeusedeveninclassesofstudentswithlit-tleornomathematicalbackground.Ithankananonymousrefereeforsuggestingthispossibili-ty.17.Iwasparticularlypleasedtoseetheseresponses,astheypartiallyalleviatedmyfearthatthestu-dentsmightbecomedependentonsymboliccomputation.Seefootnote15,infra.Summer1998245\n18.MapleappearstoperformmuchbetterinaWindows®environment,andIhaveexperiencedsub-stantialimprovementsonaMacintosh®equippedwithaPowerPCmicroprocessor.REFERENCESBinger,B.R.,andE.Hoffman.1988.Microeconomicswithcalculus.NewYork:HarperCollins.Browning,E.,andJ.Browning.1992.Microeconomictheoryandapplications.4thed.NewYork:HarperCollins.Char,B.W.,etal.1993.Firstleaves:AtutorialintroductiontoMaple.NewYork:Springer-Verlag.Gould,J.P.,andE.P.Lazear.1989.Microeconomictheory.6thed.Homewood,Ill.:RichardD.Irwin.Heck,A.1993.IntroductiontoMaple.NewYork:Springer-Verlag.Henderson,J.M.,andR.E.Quandt.1980.Microeconomictheory:Amathematicalapproach.NewYork:McGraw-Hill.Karian,Z.A.,ed.1992a.Symboliccomputationinundergraduatemathematicseducation.Washing-ton,D.C.:TheMathematicalAssociationofAmerica.———.1992b.ProceedingsoftheDenisonconferenceontheuseofsymboliccomputationinunder-graduatemathematics.Washington,D.C.:TheMathematicalAssociationofAmerica.Katz,M.L.,andH.S.Rosen.1994.Microeconomics.2ded.BurrRidge,Ill..:Irwin.King,P.G.,andR.L.Bartlett.1990Teachingeconomicsasalaboratoryscience.JournalofEco-nomicEducation21(Spring):181–93.King,P.G.,andR.M.LaRoe.1991Thelaboratory-basedeconomicscurriculum.JournalofEco-nomicEducation22(Summer):285–92.Nicholson,W.1995.Microeconomictheory:Basicprinciplesandextensions.6thed.FortWorth,Tex.:Dryden.Pindyck,R.S.,andD.L.Rubinfeld.1992.Microeconomics.2ded.NewYork:Macmillan.Varian,H.R.,ed.1992a.EconomicandfinancialmodelingwithMathematica.NewYork:Springer-Verlag.———.1992b.Microeconomicanalysis.3ded.NewYork:W.W.Norton.———.1993.Intermediatemicroeconomics:Amodernapproach.3ded.NewYork:W.W.Norton.SOLICITATIONOFPROPOSALSFOREDITORSHIPOFTHEJOURNALOFECONOMICSANDFINANCETheAcademyofEconomicsandFinanceissolicitingproposalsforeditorshipoftheJournalofEconomicsandFinance.Theeditorshipwillbeopenin1999andthetimingofthetransitionisflexible.Expectationsarethattheuniversitywillprovidereleasedtimefortheeditor,full-timeclericalhelp,andspace.Postage,supplies,long-distancecalls,andprintingofthejournalaresuppliedbytheacademy.Theacademywillconsiderajointeditorshipbetweentwoschoolsasanacceptablealternative.InquiriesorproposalsshouldbesenttoJamesT.LindleyBox5012TheUniversityofSouthernMississippiHattiesburg,MS39406ForfurtherdetailsregardingthejournalcontactJamesT.Lindley(919)933-6235or(601)266-4637;e-mail:t.lindley@usm.edu,orRichardCebula(404)894-4399;e-mailrichard.cebula@econ.gatech.edu.246JOURNALOFECONOMICEDUCATION